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# Square|Definition & Meaning

## Definition

A **square** is a flat shape with **four** straight sides such that all the sides have the same **length** and all adjacent sides are **perpendicular** to each other (all interior angles are right angles). It is a type of **quadrilateral**, a regular **polygon**, and is closely related to the class of **parallelograms** (especially the rhombus).

A **square **is also equivalent to a **rectangle **having equal length and width. Many objects can be found in your surroundings that have a **square **shape. The checkerboard, art sheets, bread slice, picture frame, pizza box, wall clock, etc., are typical instances of this shape.

A **polygon **is a shape in geometry that has at least **three **inner angles or straight edges surrounding it. They have no **curves **and are **closed shapes **constructed of **straight lines**.

The most prevalent polygons are:

- Triangle (3-sided polygon)
- Quadrilateral (4-sided polygon)
- Pentagon (5-sided polygon)
- Hexagon (6-sided polygon)
- septagon (7-sided polygon)
- Octagon (8-sided polygon) and so forth

A **polygon **is a closed, two-dimensional form with **any number **of straight sides. To put it simply, a **square **is a polygon with **four **sides.

## Square Shape

A **square **is a four-sided **polygon **with sides that are all the same **length **and **angles **that are all exactly **90 **degrees. The **square** has a form that makes both sides **symmetrical **if it is divided down the middle by a plane. The opposite sides of a square are **equal **on either half, giving the shape of a **rectangle**.

## Difference Between Square and Rectangle

The main distinction between a square and a rectangle is that the former square has equal lengths on each of its sides, but the latter has equal lengths on each of its opposite sides.

- A square has an equal length on either side. In the case of a rectangle, only the diagonal sides are equal in length.
- A square’s diagonals are each other’s perpendicular bisectors. A rectangle’s diagonals are not the perpendicular bisector of one another.
- The area of a square is obtained by multiplying either of the sides (a x a), while the area of a rectangle is the product of the length and width of its sides (l x w).
- A square’s perimeter equals 4 x. (length of a side), while a rectangle’s perimeter is equal to 2 (length + width).

## Properties of a Square

The following is a list of a square’s most key properties:

- The
**measure**of each of the interior angles is**90°**. **360°**is the result of**adding**together all interior angles.- The
**square’s**opposite sides are**parallel**to one another, and its**four**sides are**congruent**or equal to one another. - The square’s
**diagonals**form a**90°**angle with one another. - The
**square’s**two**diagonals**are**equal**to one another. - The
**square**has four**sides**and four**vertices**. - The square is divided into
**two**identical**isosceles triangles**by its diagonal. - The square’s sides are
**shorter**than its**diagonals**in length. - Its two
**diagonals**form a straight**angle**with one another.

## Square Formulas

We are aware that a **square **is a figure with four **equal **sides. In geometry, there are three fundamental **square **formulas that are frequently employed. The first one involves computing its **area**, the second one involves determining its **perimeter**, and the third one involves calculating the **diagonal **of a square formula. Let’s take a closer look at these square formulas.

### Area of a Square

The **space **a square takes up is its **area**. Chess boards, square wall clocks, and other similar objects are some instances of square shapes. To determine the **area **that these objects occupy, we can utilize the square’s **area **formula. The area of a square is calculated by **multiplying **either of its sides with each other or simply taking the square of its side’s length.

Area of square = a$^2$

where a is the side of the **square**, is how the area of a square is calculated. Square **units **such as square centimeters, square meters, and so on are used to express it.

### Perimeter of a Square

The complete **length **of a **boundary **of a square is its **perimeter**. So, by summing the **lengths **of all the **sides**, it is possible to determine the square’s perimeter. Because a **square **has **four **sides, its **perimeter **may be determined by **adding **all four of its sides.

To determine the measurement of a square’s **boundary**, we can utilize the square’s **perimeter **formula. A **square’s **perimeter can be found by adding the **lengths **of all four sides. Therefore, the square’s perimeter equals:

Perimeter of square = s + s + s + s = 4s

Linear units like cm, m, inches, and so forth are used to express it.

### Diagonal of a Square

A **line segment **connecting any two of a square’s non-adjacent **vertices **is known as the **diagonal** of a square. The **diagonals **of the square in the next figure are AC and BD. The **lengths** of the lines AC and BD are **identical**, as you can see. A diagonal **divides **a **square **generating two **equal **right **triangles**, each of which has a **diagonal **that serves as the **hypotenuse**.

Let’s examine the method used to calculate the **square’s diagonal **formula.

Let “a” denote the side length, and “d” denotes the **diagonal **length of a **square**, using the example **square **above as an example. For the triangle, we can apply the **Pythagoras** theorem.

ADC = d$^2$ = a$^2$ + a$^2$

**Square roots** on both sides result in the following:

\[ \sqrt{d^2} = \sqrt{2a^2} \]

Consequently, the formula for the **diagonal **of a **square **is:

Diagonal of a square (d) = $\sqrt{2}$ × a

## Examples Using the Formulas of Square

### Example 1

What are the **lengths **of the sides of a **square **whose area is 25 square cm?

### Solution

The **square** has an area of 25 sq. cm provided in the question.

As we are aware that taking the **square **of the length of its side gives us the **area **of a square, therefore:

Area of square = a$^2$

As a result, when we substitute the **area **value, we get the following:

25 = a$^2$

a = $\sqrt{25}$ = $\sqrt{(5)(5)}$

= 5 cm

As a result, the square’s side **length **is **5 cm**.

### Example 2

Determine the **diagonal **of a **square **with a side of 7 cm using the square’s key characteristics.

### Solution

As we know, the **diagonal** of a **square** can be computed using the formula:

d = $\sqrt{2}$ × a

Where “a” represents the side length. Plugging in the value of the **length** of the side, we get:

d = $\sqrt{2}$ × a

= $\sqrt{2}$ × 7

= 9.89 cm

Thus, the measure of the **diagonal** of a **square** having a 7 cm side length is **9.89 cm**.

*All images/mathematical drawings were created by GeoGebra.*